GENERAL REFERENCES: Batchelor, An Introduction to Fluid Dynamics, Cam- bridge University, Cambridge, 1967; Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., Wiley, New York, 2002; Brodkey, The Phenomena of Fluid Motions, Addison-Wesley, Reading, Mass., 1967; Denn, Process Fluid Mechan- ics, Prentice-Hall, Englewood Cliffs, N.J., 1980; Landau and Lifshitz, Fluid Mechanics, 2d ed., Pergamon, 1987; Govier and Aziz, The Flow of Complex Mix- tures in Pipes, Van Nostrand Reinhold, New York, 1972, Krieger, Huntington, N.Y., 1977; Panton, Incompressible Flow, Wiley, New York, 1984; Schlichting, Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987; Shames, Mechanics of Fluids, 3d ed., McGraw-Hill, New York, 1992; Streeter, Handbook of Fluid Dynamics, McGraw-Hill, New York, 1971; Streeter and Wylie, Fluid Mechanics, 8th ed., McGraw-Hill, New York, 1985; Vennard and Street, Ele- mentary Fluid Mechanics, 5th ed., Wiley, New York, 1975; Whitaker, Introduc- tion to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981. NATURE OF FLUIDS Deformation and Stress A fluid is a substance which undergoes continuous deformation when subjected to a shear stress. Figure 6-1 illustrates this concept. A fluid is bounded by two large parallel plates, of area A, separated by a small distance H. The bottom plate is held fixed. Application of a force F to the upper plate causes it to move at a velocity U. The fluid continues to deform as long as the force is applied, unlike a solid, which would undergo only a finite deformation. The force is directly proportional to the area of the plate; the shear stress is τ = F/A. Within the fluid, a linear velocity profile u = Uy/H is established; due to the no-slip condition, the fluid bounding the lower plate has zero velocity and the fluid bounding the upper plate moves at the plate velocity U. The velocity gradient γ˙ = du/dy is called the shear rate for this flow. Shear rates are usually reported in units of reciprocal seconds. The flow in Fig. 6-1 is a simple shear flow. Viscosity The ratio of shear stress to shear rate is the viscosity, µ. µ = (6-1) The SI units of viscosity are kg/(m ⋅ s) or Pa ⋅ s (pascal second). The cgs unit for viscosity is the poise; 1 Pa ⋅ s equals 10 poise or 1000 cen- tipoise (cP) or 0.672 lbm/(ft ⋅ s). The terms absolute viscosity and shear viscosity are synonymous with the viscosity as used in Eq. (6-1). Kinematic viscosity ν ϵ µ/ρ is the ratio of viscosity to density. The SI units of kinematic viscosity are m2 /s. The cgs stoke is 1 cm2 /s. Rheology In general, fluid flow patterns are more complex than the one shown in Fig. 6-1, as is the relationship between fluid defor- mation and stress. Rheology is the discipline of fluid mechanics which studies this relationship. One goal of rheology is to obtain constitu- tive equations by which stresses may be computed from deformation rates. For simplicity, fluids may be classified into rheological types in reference to the simple shear flow of Fig. 6-1. Complete definitions require extension to multidimensional flow. For more information, several good references are available, including Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977); Metzner (“Flow of Non-Newtonian Fluids” in Streeter, Handbook of Fluid Dynamics, McGraw-Hill, New York, 1971); and Skelland (Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). τ ᎏ γ˙ Fluids without any solidlike elastic behavior do not undergo any reverse deformation when shear stress is removed, and are called purely viscous fluids. The shear stress depends only on the rate of deformation, and not on the extent of deformation (strain). Those which exhibit both viscous and elastic properties are called viscoelas- tic fluids. Purely viscous fluids are further classified into time-independent and time-dependent fluids. For time-independent fluids, the shear stress depends only on the instantaneous shear rate. The shear stress for time-dependent fluids depends on the past history of the rate of deformation, as a result of structure or orientation buildup or break- down during deformation. A rheogram is a plot of shear stress versus shear rate for a fluid in simple shear flow, such as that in Fig. 6-1. Rheograms for several types of time-independent fluids are shown in Fig. 6-2. The Newtonian fluid rheogram is a straight line passing through the origin. The slope of the line is the viscosity. For a Newtonian fluid, the viscosity is inde- pendent of shear rate, and may depend only on temperature and per- haps pressure. By far, the Newtonian fluid is the largest class of fluid of engineering importance. Gases and low molecular weight liquids are generally Newtonian. Newton’s law of viscosity is a rearrangement of Eq. (6-1) in which the viscosity is a constant: τ = µγ˙ = µ (6-2) All fluids for which the viscosity varies with shear rate are non- Newtonian fluids. For non-Newtonian fluids the viscosity, defined as the ratio of shear stress to shear rate, is often called the apparent viscosity to emphasize the distinction from Newtonian behavior. Purely viscous, time-independent fluids, for which the apparent vis- cosity may be expressed as a function of shear rate, are called gener- alized Newtonian fluids. Non-Newtonian fluids include those for which a finite stress τy is required before continuous deformation occurs; these are called yield-stress materials. The Bingham plastic fluid is the simplest yield-stress material; its rheogram has a constant slope µ∞, called the infinite shear viscosity. τ = τy + µ∞γ˙ (6-3) Highly concentrated suspensions of fine solid particles frequently exhibit Bingham plastic behavior. Shear-thinning fluids are those for which the slope of the rheogram decreases with increasing shear rate. These fluids have also been called pseudoplastic, but this terminology is outdated and dis- couraged. Many polymer melts and solutions, as well as some solids suspensions, are shear-thinning. Shear-thinning fluids without yield stresses typically obey a power law model over a range of shear rates. τ = Kγ˙n (6-4) The apparent viscosity is µ = Kγ˙n − 1 (6-5) du ᎏ dy 6-4 FLUID AND PARTICLE DYNAMICS FLUID DYNAMICS y x H V F A FIG. 6-1 Deformation of a fluid subjected to a shear stress. Shear rate |du/dy| Shearstressτ τy nainotweN citsalp mahgniB citsalpo duesP tnataliD FIG. 6-2 Shear diagrams.

The factor K is the consistency index or power law coefficient, and n is the power law exponent. The exponent n is dimensionless, while K is in units of kg/(m ⋅ s2 − n ). For shear-thinning fluids, n < 1. The power law model typically provides a good fit to data over a range of one to two orders of magnitude in shear rate; behavior at very low and very high shear rates is often Newtonian. Shear-thinning power law fluids with yield stresses are sometimes called Herschel-Bulkley fluids. Numerous other rheological model equations for shear-thinning fluids are in common use. Dilatant, or shear-thickening, fluids show increasing viscosity with increasing shear rate. Over a limited range of shear rate, they may be described by the power law model with n > 1. Dilatancy is rare, observed only in certain concentration ranges in some particle sus- pensions (Govier and Aziz, pp. 33–34). Extensive discussions of dila- tant suspensions, together with a listing of dilatant systems, are given by Green and Griskey (Trans. Soc. Rheol, 12[1], 13–25 [1968]); Griskey and Green (AIChE J., 17, 725–728 [1971]); and Bauer and Collins (“Thixotropy and Dilatancy,” in Eirich, Rheology, vol. 4, Aca- demic, New York, 1967). Time-dependent fluids are those for which structural rearrange- ments occur during deformation at a rate too slow to maintain equi- librium configurations. As a result, shear stress changes with duration of shear. Thixotropic fluids, such as mayonnaise, clay suspensions used as drilling muds, and some paints and inks, show decreasing shear stress with time at constant shear rate. A detailed description of thixotropic behavior and a list of thixotropic systems is found in Bauer and Collins (ibid.). Rheopectic behavior is the opposite of thixotropy. Shear stress increases with time at constant shear rate. Rheopectic behavior has been observed in bentonite sols, vanadium pentoxide sols, and gyp- sum suspensions in water (Bauer and Collins, ibid.) as well as in some polyester solutions (Steg and Katz, J. Appl. Polym. Sci., 9, 3, 177 [1965]). Viscoelastic fluids exhibit elastic recovery from deformation when stress is removed. Polymeric liquids comprise the largest group of flu- ids in this class. A property of viscoelastic fluids is the relaxation time, which is a measure of the time required for elastic effects to decay. Viscoelastic effects may be important with sudden changes in rates of deformation, as in flow startup and stop, rapidly oscillating flows, or as a fluid passes through sudden expansions or contractions where accel- erations occur. In many fully developed flows where such effects are absent, viscoelastic fluids behave as if they were purely viscous. In vis- coelastic flows, normal stresses perpendicular to the direction of shear are different from those in the parallel direction. These give rise to such behaviors as the Weissenberg effect, in which fluid climbs up a shaft rotating in the fluid, and die swell, where a stream of fluid issu- ing from a tube may expand to two or more times the tube diameter. A parameter indicating whether viscoelastic effects are important is the Deborah number, which is the ratio of the characteristic relax- ation time of the fluid to the characteristic time scale of the flow. For small Deborah numbers, the relaxation is fast compared to the char- acteristic time of the flow, and the fluid behavior is purely viscous. For very large Deborah numbers, the behavior closely resembles that of an elastic solid. Analysis of viscoelastic flows is very difficult. Simple constitutive equations are unable to describe all the material behavior exhibited by viscoelastic fluids even in geometrically simple flows. More complex constitutive equations may be more accurate, but become exceedingly difficult to apply, especially for complex geometries, even with advanced numerical methods. For good discussions of viscoelastic fluid behavior, including various types of constitutive equations, see Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, vol. 2: Kinetic Theory, Wiley, New York, 1977); Middleman (The Flow of High Polymers, Interscience (Wiley) New York, 1968); or Astarita and Marrucci (Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974). Polymer processing is the field which depends most on the flow of non-Newtonian fluids. Several excellent texts are available, including Middleman (Fundamentals of Polymer Processing, McGraw-Hill, New York, 1977) and Tadmor and Gogos (Principles of Polymer Processing, Wiley, New York, 1979). There is a wide variety of instruments for measurement of Newto- nian viscosity, as well as rheological properties of non-Newtonian flu- ids. They are described in Van Wazer, Lyons, Kim, and Colwell (Viscosity and Flow Measurement, Interscience, New York, 1963); Coleman, Markowitz, and Noll (Viscometric Flows of Non-Newtonian Fluids, Springer-Verlag, Berlin, 1966); Dealy and Wissbrun (Melt Rheology and Its Role in Plastics Processing, Van Nostrand Reinhold, 1990). Measurement of rheological behavior requires well-characterized flows. Such rheometric flows are thoroughly discussed by Astarita and Marrucci (Principles of Non-Newtonian Fluid Mechanics, McGraw- Hill, New York, 1974). KINEMATICS OF FLUID FLOW Velocity The term kinematics refers to the quantitative descrip- tion of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vec- tor quantity, with three cartesian components vx, vy, and vz. The veloc- ity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. Compressible and Incompressible Flow An incompressible flow is one in which the density of the fluid is constant or nearly con- stant. Liquid flows are normally treated as incompressible, except in the context of hydraulic transients (see following). Compressible flu- ids, such as gases, may undergo incompressible flow if pressure and/or temperature changes are small enough to render density changes insignificant. Frequently, compressible flows are regarded as flows in which the density varies by more than 5 to 10 percent. Streamlines, Pathlines, and Streaklines These are curves in a flow field which provide insight into the flow pattern. Streamlines are tangent at every point to the local instantaneous velocity vector. A pathline is the path followed by a material element of fluid; it coin- cides with a streamline if the flow is steady. In unsteady flow the path- lines generally do not coincide with streamlines. Streaklines are curves on which are found all the material particles which passed through a particular point in space at some earlier time. For example, a streakline is revealed by releasing smoke or dye at a point in a flow field. For steady flows, streamlines, pathlines, and streaklines are indistinguishable. In two-dimensional incompressible flows, stream- lines are contours of the stream function. One-dimensional Flow Many flows of great practical impor- tance, such as those in pipes and channels, are treated as one- dimensional flows. There is a single direction called the flow direction; velocity components perpendicular to this direction are either zero or considered unimportant. Variations of quantities such as velocity, pressure, density, and temperature are considered only in the flow direction. The fundamental conservation equations of fluid mechanics are greatly simplified for one-dimensional flows. A broader category of one-dimensional flow is one where there is only one nonzero veloc- ity component, which depends on only one coordinate direction, and this coordinate direction may or may not be the same as the flow direction. Rate of Deformation Tensor For general three-dimensional flows, where all three velocity components may be important and may vary in all three coordinate directions, the concept of deformation previously introduced must be generalized. The rate of deformation tensor Dij has nine components. In Cartesian coordinates, Dij = ΂ + ΃ (6-6) where the subscripts i and j refer to the three coordinate directions. Some authors define the deformation rate tensor as one-half of that given by Eq. (6-6). Vorticity The relative motion between two points in a fluid can be decomposed into three components: rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilata- tion refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described by a ten- sor ωij = ∂vi /∂xj − ∂vj /∂xi. The vector of vorticity given by one-half the ∂vj ᎏ ∂xi ∂vi ᎏ ∂xj FLUID DYNAMICS 6-5

curl of the velocity vector is another measure of rotation. In two- dimensional flow in the x-y plane, the vorticity ω is given by ω = ΂ − ΃ (6-7) Here ω is the magnitude of the vorticity vector, which is directed along the z axis. An irrotational flow is one with zero vorticity. Irro- tational flows have been widely studied because of their useful math- ematical properties and applicability to flow regions where viscous effects may be neglected. Such flows without viscous effects are called inviscid flows. Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fluid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU ρ/µ where L is a characteristic length. Below a critical value of Re the flow is lam- inar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is deter- mined experimentally. CONSERVATION EQUATIONS Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con- servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequal- ity (second law of thermodynamics) have occasional use. The conser- vation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential conservation equations, respectively. These are often called macroscopic and microscopic balance equa- tions. Macroscopic Equations An arbitrary control volume of finite size Va is bounded by a surface of area Aa with an outwardly directed unit normal vector n. The control volume is not necessarily fixed in space. Its boundary moves with velocity w. The fluid velocity is v. Fig- ure 6-3 shows the arbitrary control volume. Mass Balance Applied to the control volume, the principle of conservation of mass may be written as (Whitaker, Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981) ͵Va ρ dV + ͵Aa ρ(v − w) ⋅ n dA = 0 (6-8) This equation is also known as the continuity equation. d ᎏ dt ∂vx ᎏ ∂y ∂vy ᎏ ∂x 1 ᎏ 2 Simplified forms of Eq. (6-8) apply to special cases frequently found in practice. For a control volume fixed in space with one inlet of area A1 through which an incompressible fluid enters the control vol- ume at an average velocity V1, and one outlet of area A2 through which fluid leaves at an average velocity V2, as shown in Fig. 6-4, the conti- nuity equation becomes V1 A1 = V2 A2 (6-9) The average velocity across a surface is given by V = (1/A) ͵A v dA where v is the local velocity component perpendicular to the inlet sur- face. The volumetric flow rate Q is the product of average velocity and the cross-sectional area, Q = VA. The average mass velocity is G = ρV. For steady flows through fixed control volumes with multiple inlets and/or outlets, conservation of mass requires that the sum of inlet mass flow rates equals the sum of outlet mass flow rates. For incompressible flows through fixed control volumes, the sum of inlet flow rates (mass or volumetric) equals the sum of exit flow rates, whether the flow is steady or unsteady. Momentum Balance Since momentum is a vector quantity, the momentum balance is a vector equation. Where gravity is the only body force acting on the fluid, the linear momentum principle, applied to the arbitrary control volume of Fig. 6-3, results in the fol- lowing expression (Whitaker, ibid.). ͵Va ρv dV + ͵Aa ρv(v − w) ⋅ n dA = ͵Va ρg dV + ͵Aa tn dA (6-10) Here g is the gravity vector and tn is the force per unit area exerted by the surroundings on the fluid in the control volume. The integrand of the area integral on the left-hand side of Eq. (6-10) is nonzero only on the entrance and exit portions of the control volume boundary. For the special case of steady flow at a mass flow rate ˙m through a control volume fixed in space with one inlet and one outlet (Fig. 6-4), with the inlet and outlet velocity vectors perpendicular to planar inlet and out- let surfaces, giving average velocity vectors V1 and V2, the momentum equation becomes ˙m(β2V2 − β1V1) = −p1A1 − p2A2 + F + Mg (6-11) where M is the total mass of fluid in the control volume. The factor β arises from the averaging of the velocity across the area of the inlet or outlet surface. It is the ratio of the area average of the square of veloc- ity magnitude to the square of the area average velocity magnitude. For a uniform velocity, β = 1. For turbulent flow, β is nearly unity, while for laminar pipe flow with a parabolic velocity profile, β = 4/3. The vectors A1 and A2 have magnitude equal to the areas of the inlet and outlet surfaces, respectively, and are outwardly directed normal to the surfaces. The vector F is the force exerted on the fluid by the non- flow boundaries of the control volume. It is also assumed that the stress vector tn is normal to the inlet and outlet surfaces, and that its magnitude may be approximated by the pressure p. Equation (6-11) may be generalized to multiple inlets and/or outlets. In such cases, the mass flow rates for all the inlets and outlets are not equal. A distinct flow rate ˙mi applies to each inlet or outlet i. To generalize the equa- tion, ؊pA terms for each inlet and outlet, − ˙mβV terms for each inlet, and ˙mβV terms for each outlet are included. d ᎏ dt 6-6 FLUID AND PARTICLE DYNAMICS Volume Va Area Aa n outwardly directed unit normal vector w boundary velocity v fluid velocity FIG. 6-3 Arbitrary control volume for application of conservation equations. FIG. 6-4 Fixed control volume with one inlet and one outlet. V1 V2 1 2

Balance equations for angular momentum, or moment of momen- tum, may also be written. They are used less frequently than the linear momentum equations. See Whitaker (Introduction to Fluid Mechan- ics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981) or Shames (Mechanics of Fluids, 3d ed., McGraw-Hill, New York, 1992). Total Energy Balance The total energy balance derives from the first law of thermodynamics. Applied to the arbitrary control vol- ume of Fig. 6-3, it leads to an equation for the rate of change of the sum of internal, kinetic, and gravitational potential energy. In this equation, u is the internal energy per unit mass, v is the magnitude of the velocity vector v, z is elevation, g is the gravitational acceleration, and q is the heat flux vector: ͵Va ρ΂u + + gz΃dV + ͵Aa ρ΂u + + gz΃(v − w) ⋅ n dA = ͵Aa (v ⋅ tn) dA − ͵Aa (q ⋅ n) dA (6-12) The first integral on the right-hand side is the rate of work done on the fluid in the control volume by forces at the boundary. It includes both work done by moving solid boundaries and work done at flow entrances and exits. The work done by moving solid boundaries also includes that by such surfaces as pump impellers; this work is called shaft work; its rate is ˙WS. A useful simplification of the total energy equation applies to a par- ticular set of assumptions. These are a control volume with fixed solid boundaries, except for those producing shaft work, steady state condi- tions, and mass flow at a rate ˙m through a single planar entrance and a single planar exit (Fig. 6-4), to which the velocity vectors are per- pendicular. As with Eq. (6-11), it is assumed that the stress vector tn is normal to the entrance and exit surfaces and may be approximated by the pressure p. The equivalent pressure, p + ρgz, is assumed to be uniform across the entrance and exit. The average velocity at the entrance and exit surfaces is denoted by V. Subscripts 1 and 2 denote the entrance and exit, respectively. h1 + α1 + gz1 = h2 + α2 + gz2 − δQ − δWS (6-13) Here, h is the enthalpy per unit mass, h = u + p/ρ. The shaft work per unit of mass flowing through the control volume is δWS = ˙Ws /˙m. Sim- ilarly, δQ is the heat input per unit of mass. The factor α is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, α = 1. In turbu- lent flow, α is usually assumed to equal unity; in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circular pipe with a para- bolic velocity profile, α = 2. Mechanical Energy Balance, Bernoulli Equation A balance equation for the sum of kinetic and potential energy may be obtained from the momentum balance by forming the scalar product with the velocity vector. The resulting equation, called the mechanical energy balance, contains a term accounting for the dissipation of mechanical energy into thermal energy by viscous forces. The mechanical energy equation is also derivable from the total energy equation in a way that reveals the relationship between the dissipation and entropy genera- tion. The macroscopic mechanical energy balance for the arbitrary control volume of Fig. 6-3 may be written, with p = thermodynamic pressure, as ͵Va ρ΂ + gz΃dV + ͵Aa ρ΂ + gz΃(v − w) ⋅ n dA = ͵Va p ١ ⋅ v dV + ͵Aa (v ⋅ tn) dA − ͵Va Φ dV (6-14) The last term is the rate of viscous energy dissipation to internal energy, ˙Ev = ͵Va Φ dV, also called the rate of viscous losses. These losses are the origin of frictional pressure drop in fluid flow. Whitaker and Bird, Stewart, and Lightfoot provide expressions for the dissipa- tion function Φ for Newtonian fluids in terms of the local velocity gra- dients. However, when using macroscopic balance equations the local velocity field within the control volume is usually unknown. For such v2 ᎏ 2 v2 ᎏ 2 d ᎏ dt V2 2 ᎏ 2 V2 1 ᎏ 2 v2 ᎏ 2 v2 ᎏ 2 d ᎏ dt cases additional information, which may come from empirical correla- tions, is needed. For the same special conditions as for Eq. (6-13), the mechanical energy equation is reduced to α1 + gz1 + δWS = α2 + gz2 + ͵ p 2 p1 + lv (6-15) Here lv = ˙Ev /˙m is the energy dissipation per unit mass. This equation has been called the engineering Bernoulli equation. For an incompressible flow, Eq. (6-15) becomes + α1 + gz1 + δWS = + α2 + gz2 + lv (6-16) The Bernoulli equation can be written for incompressible, inviscid flow along a streamline, where no shaft work is done. + + gz1 = + + gz2 (6-17) Unlike the momentum equation (Eq. [6-11]), the Bernoulli equation is not easily generalized to multiple inlets or outlets. Microscopic Balance Equations Partial differential balance equations express the conservation principles at a point in space. Equations for mass, momentum, total energy, and mechanical energy may be found in Whitaker (ibid.), Bird, Stewart, and Lightfoot (Trans- port Phenomena, Wiley, New York, 1960), and Slattery (Momentum, Heat and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981), for example. These references also present the equations in other useful coordinate systems besides the cartesian system. The coordinate systems are fixed in inertial reference frames. The two most used equations, for mass and momentum, are presented here. Mass Balance, Continuity Equation The continuity equation, expressing conservation of mass, is written in cartesian coordinates as + + + = 0 (6-18) In terms of the substantial derivative, D/Dt, ϵ + vx + vy + vz = −ρ΂ + + ΃ (6-19) The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may be expressed as = −ρ∇ ⋅ v (6-20) For incompressible flow, ∇ ⋅ v = + + = 0 (6-21) Stress Tensor The stress tensor is needed to completely describe the stress state for microscopic momentum balances in multidimen- sional flows. The components of the stress tensor σij give the force in the j direction on a plane perpendicular to the i direction, using a sign convention defining a positive stress as one where the fluid with the greater i coordinate value exerts a force in the positive i direction on the fluid with the lesser i coordinate. Several references in fluid mechanics and continuum mechanics provide discussions, to various levels of detail, of stress in a fluid (Denn; Bird, Stewart, and Lightfoot; Schlichting; Fung [A First Course in Continuum Mechanics, 2d. ed., Prentice-Hall, Englewood Cliffs, N.J., 1977]; Truesdell and Toupin [in Flügge, Handbuch der Physik, vol. 3/1, Springer-Verlag, Berlin, 1960]; Slattery [Momentum, Energy and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981]). The stress has an isotropic contribution due to fluid pressure and dilatation, and a deviatoric contribution due to viscous deformation effects. The deviatoric contribution for a Newtonian fluid is the three- dimensional generalization of Eq. (6-2): τij = µDij (6-22) ∂vz ᎏ ∂z ∂vy ᎏ ∂y ∂vx ᎏ ∂x Dρ ᎏ Dt ∂vz ᎏ ∂z ∂vy ᎏ ∂y ∂vx ᎏ ∂x ∂ρ ᎏ ∂z ∂ρ ᎏ ∂y ∂ρ ᎏ ∂x ∂ρ ᎏ ∂t Dρ ᎏ Dt ∂ρvz ᎏ ∂z ∂ρvy ᎏ ∂y ∂ρvx ᎏ ∂x ∂ρ ᎏ ∂t V2 2 ᎏ 2 p2 ᎏ ρ V2 1 ᎏ 2 p1 ᎏ ρ V2 2 ᎏ 2 p2 ᎏ ρ V2 1 ᎏ 2 p1 ᎏ ρ dp ᎏ ρ V2 2 ᎏ 2 V2 1 ᎏ 2 FLUID DYNAMICS 6-7

The total stress is σij = (−p + λ∇ ⋅ v)δij + τij (6-23) The identity tensor δij is zero for i ≠ j and unity for i = j. The coefficient λ is a material property related to the bulk viscosity, κ = λ + 2µ/3. There is considerable uncertainty about the value of κ. Traditionally, Stokes’ hypothesis, κ = 0, has been invoked, but the validity of this hypothesis is doubtful (Slattery, ibid.). For incompressible flow, the value of bulk viscosity is immaterial as Eq. (6-23) reduces to σij = −pδij + τij (6-24) Similar generalizations to multidimensional flow are necessary for non-Newtonian constitutive equations. Cauchy Momentum and Navier-Stokes Equations The dif- ferential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fluid mechanics texts (e.g., Slattery [ibid.]; Denn; Whitaker; and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) leads to the Navier-Stokes equations, whose three Cartesian components are ρ΂ + vx + vy + vz ΃ = − + µ΂ + + ΃+ ρgx (6-25) ρ΂ + vx + vy + vz ΃ = − + µ΂ + + ΃+ ρgy (6-26) ρ΂ + vx + vy + vz ΃ = − + µ΂ + + ΃+ ρgz (6-27) In vector notation, ρ = + (v ⋅ ∇)v = −∇p + µ∇2 v + ρg (6-28) The pressure and gravity terms may be combined by replacing the pressure p by the equivalent pressure P = p + ρgz. The left-hand side terms of the Navier-Stokes equations are the inertial terms, while the terms including viscosity µ are the viscous terms. Limiting cases under which the Navier-Stokes equations may be simplified include creeping flows in which the inertial terms are neglected, potential flows (inviscid or irrotational flows) in which the viscous terms are neglected, and boundary layer and lubrication flows in which cer- tain terms are neglected based on scaling arguments. Creeping flows are described by Happel and Brenner (Low Reynolds Number Hydro- dynamics, Prentice-Hall, Englewood Cliffs, N.J., 1965); potential flows by Lamb (Hydrodynamics, 6th ed., Dover, New York, 1945) and Milne-Thompson (Theoretical Hydrodynamics, 5th ed., Macmillan, New York, 1968); boundary layer theory by Schlichting (Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987); and lubrica- tion theory by Batchelor (An Introduction to Fluid Dynamics, Cambridge University, Cambridge, 1967) and Denn (Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1980). Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure boundary condition and two velocity boundary conditions (for each velocity com- ponent) to completely specify the solution. The no slip condition, which requires that the fluid velocity equal the velocity of any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather ∂v ᎏ ∂t Dv ᎏ Dt ∂2 vz ᎏ ∂z2 ∂2 vz ᎏ ∂y2 ∂2 vz ᎏ ∂x2 ∂p ᎏ ∂z ∂vz ᎏ ∂z ∂vz ᎏ ∂y ∂vz ᎏ ∂x ∂vz ᎏ ∂t ∂2 vy ᎏ ∂z2 ∂2 vy ᎏ ∂y2 ∂2 vy ᎏ ∂x2 ∂p ᎏ ∂y ∂vy ᎏ ∂z ∂vy ᎏ ∂y ∂vy ᎏ ∂x ∂vy ᎏ ∂t ∂2 vx ᎏ ∂z2 ∂2 vx ᎏ ∂y2 ∂2 vx ᎏ ∂x2 ∂p ᎏ ∂x ∂vx ᎏ ∂z ∂vx ᎏ ∂y ∂vx ᎏ ∂x ∂vx ᎏ ∂t than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-Stokes equations, Dirichlet and Neumann, or essential and natural, boundary condi- tions may be satisfied by different means. Fluid statics, discussed in Sec. 10 of the Handbook in reference to pressure measurement, is the branch of fluid mechanics in which the fluid velocity is either zero or is uniform and constant relative to an inertial reference frame. With velocity gradients equal to zero, the momentum equation reduces to a simple expression for the pressure field, ∇p = ρg. Letting z be directed vertically upward, so that gz = −g where g is the gravitational acceleration (9.806 m2 /s), the pressure field is given by dp/dz = −ρg (6-29) This equation applies to any incompressible or compressible static fluid. For an incompressible liquid, pressure varies linearly with depth. For compressible gases, p is obtained by integration account- ing for the variation of ρ with z. The force exerted on a submerged planar surface of area A is given by F = pc A where pc is the pressure at the geometrical centroid of the surface. The center of pressure, the point of application of the net force, is always lower than the centroid. For details see, for example, Shames, where may also be found discussion of forces on curved surfaces, buoyancy, and stability of floating bodies. Examples Four examples follow, illustrating the application of the conservation equations to obtain useful information about fluid flows. Example 1: Force Exerted on a Reducing Bend An incompress- ible fluid flows through a reducing elbow (Fig. 6-5) situated in a horizontal plane. The inlet velocity V1 is given and the pressures p1 and p2 are measured. Selecting the inlet and outlet surfaces 1 and 2 as shown, the continuity equation Eq. (6-9) can be used to find the exit velocity V2 = V1A1/A2. The mass flow rate is obtained by ˙m = ρV1A1. Assume that the velocity profile is nearly uniform so that β is approximately unity. The force exerted on the fluid by the bend has x and y components; these can be found from Eq. (6-11). The x component gives Fx = ˙m(V2x − V1x) + p1A1x + p2 A2x while the y component gives Fy = ˙m(V2y − V1y) + p1 A1y + p2 A2y The velocity components are V1x = V1, V1y = 0, V2x = V2 cos θ, and V2y = V2 sin θ. The area vector components are A1x = −A1, A1y = 0, A2x = A2 cos θ, and A2y = A2 sin θ. Therefore, the force components may be calculated from Fx = ˙m(V2 cos θ − V1) − p1A1 + p2A2 cos θ Fy = ˙mV2 sin θ + p2A2 sin θ The force acting on the fluid is F; the equal and opposite force exerted by the fluid on the bend is ؊F. 6-8 FLUID AND PARTICLE DYNAMICS V1 V2 F θ y x FIG. 6-5 Force at a reducing bend. F is the force exerted by the bend on the fluid. The force exerted by the fluid on the bend is ؊F.

Example 2: Simplified Ejector Figure 6-6 shows a very simplified sketch of an ejector, a device that uses a high velocity primary fluid to pump another (secondary) fluid. The continuity and momentum equations may be applied on the control volume with inlet and outlet surfaces 1 and 2 as indicated in the figure. The cross-sectional area is uniform, A1 = A2 = A. Let the mass flow rates and velocities of the primary and secondary fluids be ˙mp, ˙ms, Vp and Vs. Assume for simplicity that the density is uniform. Conservation of mass gives m˙2 = ˙mp + ˙ms. The exit velocity is V2 = ˙m2 /(ρA). The principle momentum exchange in the ejector occurs between the two fluids. Relative to this exchange, the force exerted by the walls of the device are found to be small. Therefore, the force term F is neglected from the momentum equation. Written in the flow direction, assuming uniform velocity profiles, and using the extension of Eq. (6-11) for multiple inlets, it gives the pressure rise developed by the device: (p2 − p1)A = (m˙ p + ˙ms)V2 − ˙mpVp − ˙msVs Application of the momentum equation to ejectors of other types is discussed in Lapple (Fluid and Particle Dynamics, University of Delaware, Newark, 1951) and in Sec. 10 of the Handbook. Example 3: Venturi Flowmeter An incompressible fluid flows through the venturi flowmeter in Fig. 6-7. An equation is needed to relate the flow rate Q to the pressure drop measured by the manometer. This problem can be solved using the mechanical energy balance. In a well-made venturi, viscous losses are negligible, the pressure drop is entirely the result of acceleration into the throat, and the flow rate predicted neglecting losses is quite accurate. The inlet area is A and the throat area is a. With control surfaces at 1 and 2 as shown in the figure, Eq. (6-17) in the absence of losses and shaft work gives + = + The continuity equation gives V2 = V1A/a, and V1 = Q/A. The pressure drop mea- sured by the manometer is p1 − p2 = (ρm − ρ)g∆z. Substituting these relations into the energy balance and rearranging, the desired expression for the flow rate is found. Q = Ί๶ Example 4: Plane Poiseuille Flow An incompressible Newtonian fluid flows at a steady rate in the x direction between two very large flat plates, as shown in Fig. 6-8. The flow is laminar. The velocity profile is to be found. This example is found in most fluid mechanics textbooks; the solution presented here closely follows Denn. 2(ρm − ρ)g∆z ᎏᎏ ρ[(A/a)2 − 1] 1 ᎏ A V2 2 ᎏ 2 p2 ᎏ ρ V2 1 ᎏ 2 p1 ᎏ ρ This problem requires use of the microscopic balance equations because the velocity is to be determined as a function of position. The boundary conditions for this flow result from the no-slip condition. All three velocity components must be zero at the plate surfaces, y = H/2 and y = −H/2. Assume that the flow is fully developed, that is, all velocity derivatives vanish in the x direction. Since the flow field is infinite in the z direction, all velocity derivatives should be zero in the z direction. Therefore, velocity components are a function of y alone. It is also assumed that there is no flow in the z direction, so vz = 0. The continuity equation Eq. (6-21), with vz = 0 and ∂vx /∂x = 0, reduces to = 0 Since vy = 0 at y = ϮH/2, the continuity equation integrates to vy = 0. This is a direct result of the assumption of fully developed flow. The Navier-Stokes equations are greatly simplified when it is noted that vy = vz = 0 and ∂vx /∂x = ∂vx /∂z = ∂vx /∂t = 0. The three components are written in terms of the equivalent pressure P: 0 = − + µ 0 = − 0 = − The latter two equations require that P is a function only of x, and therefore ∂P/∂x = dP/dx. Inspection of the first equation shows one term which is a func- tion only of x and one which is only a function of y. This requires that both terms are constant. The pressure gradient −dP/dx is constant. The x-component equa- tion becomes = Two integrations of the x-component equation give vx = y2 + C1y + C2 where the constants of integration C1 and C2 are evaluated from the boundary conditions vx = 0 at y = ϮH/2. The result is vx = ΂− ΃΄1 − ΂ ΃ 2 ΅ This is a parabolic velocity distribution. The average velocity V = (1/H) ͵H/2 −H/2 vx dy is V = ΂− ΃ This flow is one-dimensional, as there is only one nonzero velocity component, vx, which, along with the pressure, varies in only one coordinate direction. INCOMPRESSIBLE FLOW IN PIPES AND CHANNELS Mechanical Energy Balance The mechanical energy balance, Eq. (6-16), for fully developed incompressible flow in a straight cir- cular pipe of constant diameter D reduces to + gz1 = + gz2 + lv (6-30) In terms of the equivalent pressure, P ϵ p + ρgz, P1 − P2 = ρlv (6-31) The pressure drop due to frictional losses lv is proportional to pipe length L for fully developed flow and may be denoted as the (positive) quantity ∆P ϵ P1 − P2. p2 ᎏ ρ p1 ᎏ ρ dP ᎏ dx H2 ᎏ 12µ 2y ᎏ H dP ᎏ dx H2 ᎏ 8µ dP ᎏ dx 1 ᎏ 2µ dP ᎏ dx 1 ᎏ µ d2 vx ᎏ dy2 ∂P ᎏ ∂z ∂P ᎏ ∂y ∂2 vx ᎏ ∂y2 ∂P ᎏ ∂x dvy ᎏ dy FLUID DYNAMICS 6-9 FIG. 6-6 Draft-tube ejector. ∆z 1 2 FIG. 6-7 Venturi flowmeter. y x H FIG. 6-8 Plane Poiseuille flow.

thunderbooks.files.wordpress.com

These presentations are classified and categorized, so you will always find everything clearly laid out and in context.
You are watching Kedaulatan Rakyat 30 Maret 2014 presentation right now. We are staying up to date!